Hardness of approximation, aka inapproximability.

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Definition of Projection Measure in the characterization of strong approximation Resistance in a paper by Khot et al

I'm reading a paper about Constraint Satisfaction Problems, specifically "A Characterization of Strong Approximation Resistance", Subhash Khot, Madhur Tulsiani, Pratik Worah (ECCC TR13-075). The ...
8
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1answer
172 views

A purely graph-theoretic explanation of the reduction from Unique Label Cover to Max-Cut

I am studying the Unique Games Conjecture and the famous reduction to Max-Cut of Khot et al. From their paper and elsewhere on the internet, most authors use (what to me is) an implicit equivalence ...
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39 views

Minimum vector sets span spaces cover problem

Instance: a set of vectors $V=\{\mathbf{v}_1,\mathbf{v}_2,...,\mathbf{v}_n\}$ and $m$ vector sets $V_1,V_2,...,V_m$, each of which contains multiple vectors ($V_i$ may not be a subset of $V$). In our ...
4
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1answer
111 views

Inapproximability of $(\alpha, \beta)$ bi-criteria approximation

An $(\alpha, \beta)$ bi-criteria approximation algorithm for $k$-center is defined as an algorithm that returns a solution whose value is $\beta \cdot OPT$ ($OPT$ being the optimal solution for the ...
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0answers
42 views

inapproximability of logarithic factor of indepence set [closed]

The hardness result derived using PCP theorem for Independent set suggests that there exists some absolute constant $\epsilon_0$ such that for $0< \epsilon < \epsilon_0$, it is hard to ...
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170 views
+50

Is graph coloring complete for poly-APX?

Is the graph coloring problem complete for poly-APX under C-reductions (alternatively, under AP-reductions)? For the graph coloring problem, speaking of a feasible solution means a proper coloring for ...
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0answers
46 views

How can I show that zero-one programming is not in APX?

How can I show that zero-one programming is not in APX? Vertex Cover Problem is in the APX class. So can I try a PTAS-reduction from the zero-one programming problem to Vertex Cover and show that ...
8
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2answers
164 views

Approximating #P-hard problems

Consider the classical #P-complete problem #3SAT, i.e., to count the number of valuations to make a 3CNF with $n$ variables satisfiable. I am interested in the additive approximability. Clearly, there ...
2
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164 views

Complexity of linear programming

It is known that Linear Programming (LP) is P-complete. I am interested in approximation algorithms for LP. There are numerous inapproximability results for NP optimization problems, e.g. it is ...
0
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125 views

Calculating exact/approximate solution to a formula

Suppose we have a set of variable $\mathbf{y} = \left(y_1, ..., y_n \right)$. Also consider the set of functions $g_i(y_i), 1 \leq i \leq n$. Note that $g_i()$ is dependent only on $y_i$. Consider ...
2
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1answer
103 views

Hardness of approximating chromatic number of triangle-free graphs

The chromatic number of graph, $\chi( G)$ is hard to approximate for general graphs. Are there results of hardness of approximating $\chi(G)$ for triangle-free graphs?
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Approximate c-chromatic number, each color class is P4-free (cograph)

The classic chromatic number of graph, $\chi(G)$, describes the minimum number of colors needed so that each color class is an independent set. There are many other graph coloring definitions. One of ...
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0answers
30 views

Approximation algorithm to a minimization problem with optimization function going to zero

I have a simple function to minimize, but it is not discrete: $f(x) = 2x - \alpha x - c(x)$, where $x$ is a natural number, $\alpha$ is a rational number, $0 \leq \alpha \leq 1$, and $0 \leq c(x) \leq ...
14
votes
1answer
332 views

Does PSPACE-completeness imply approximation hardness?

It is mentioned in a comment in another cstheorySE post that PSPACE-completeness imply APX-hardness. Can anyone please explain/share a reference for it? Is this "tight"? (i.e., are there ...
7
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2answers
217 views

Planted Clique in G(n,p), varying p

In the planted clique problem, one must recover a $k$-clique planted in an Erdos-Renyi random graph $G(n,p)$. This has mostly been looked at for $p=\frac{1}{2}$, in which case it is known to be ...
2
votes
2answers
150 views

Set packing with maximum coverage objective

We are given a universe $\mathcal{U}=\{e_1,..,e_n\}$ and a set of subsets $\mathcal{S}=\{s_1,s_2,...,s_m\}\subseteq 2^\mathcal{U}$. Set-Packing asks how many disjoint sets we can pack, and is defined ...
4
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1answer
59 views

Bellman principle and approximability

Does anybody know if a combinatorial optimzation problem that enjoys the Bellman's optimality principle can in automatic way be approximated?
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3answers
165 views

Comparing graphs

I am looking for a kick in the right direction. What i am trying to do is to compute "similarity" between two graphs, where I define "similarity" as the number of shared paths. example: ...
10
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1answer
193 views

Almost always almost right

I am looking for a complexity class that relates to APX as BPP relates to P. I have already asked the same question here, but perhaps TCS would be a more fruitful location for answers. The reason for ...
3
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2answers
197 views

Variants of Densest Subgraph Problems

Given a graph $G$, and a vertex-induced subgraph $H$ of $G$, there are two superficially-similar definitions ways to define the density of $H$: (1) Average degree of vertices in $H$ (2) What I will ...
14
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2answers
255 views

Approximation in subexponental time

There are studies about approximation algorithms for NP complete problems in Polynomial time and exact algorithms in exponential time. Are there studies about approximation algorithms for NP complete ...
2
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0answers
42 views

Approximating BLEDP on restricted graph classes

In the edge-disjoint paths (EDP) problem, we are given an undirected graph $G$, and a set $\{ (s_i,t_i) \mid 1 \leq i \leq k \}$ of $k$ source-sink pairs. The objective is to maximize the number of ...
3
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1answer
84 views

Approximation algorithms for multicut for special classes of graphs

The multicut problem is the following. Given a graph $G=(V,E)$ with edge costs $c_e$ for edge $e$ and a set of $k$ terminal pairs $\{(s_1,t_1),\ldots,(s_k,t_k)\}$, the objective is to find a set of ...
2
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0answers
112 views

The non-metric k-median problem

It is well-known that the non-metric $k$-median problem cannot be approximated better than $O(\log(n))$ (by a gap preserving reduction from the set cover problem). Is there any logarithmic ...
10
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2answers
418 views

Reference Request: Asymptotic hardness of $hk$ coloring $k$-colorable graphs

I heard of a result in approximate graph coloring, but cannot find the source. The result is: For every constant $h$ there exists a sufficiently large $k$ such that coloring a $k$-colorable graph ...
2
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2answers
264 views

Multi prover, verifier games and PCP theorem

This question came up while I was going through Siu On Chan's paper on Approximation Resistance. My question is not really related to the paper though. I also guess that this is more of a reference ...
6
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1answer
207 views

Complexity of finding large grid minors

What is the complexity of finding the largest $k\times k$ grid graph that is a minor of a given graph $G$? It is FPT in $k$, and it seems likely to be NP-hard (or NP-complete in a decision version ...
7
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4answers
782 views

Hardest problems to approximate

Under some assumptions, it is hard to approximate MAX-CLIQUE within a factor $n^{1-\epsilon}$ for any $\epsilon >0$. Are there any other problems that are known to be equally hard to approximate? ...
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1answer
104 views

number of PCP queries

we know from the PCP theorem that $PCP[O(log(n)),O(1)]=NP$,what if we choose specific number of queries will the theorem hold ?
12
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1answer
295 views

Smoothed analysis of approximation algorithms

Smoothed analysis has been applied many times to understand the runtime of exact algorithms for many problems like linear programming and k-means. There are fairly general results in this realm, for ...
6
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4answers
655 views

How to start studying topics Hardness of approximation and PCP's

Recently I have done an introductory course on complexity theory ( which covered 90% of sipser text book). Now I would like to study the topics Hardness of approximation and PCP's. Can you please ...
8
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234 views

Hitting sets with a subfamily

Let $F$ be a family of $d$-element subsets of a finite universe $U$ of objects. A family $H$ of $k$-element subsets of $U$, with $1 \le k < d$, is a $(k,d)$-hitting-set of $F$ if for each $V \in F$ ...
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Approximation for accumulative set cover

Let $S_1,\ldots,S_m\subseteq U$ be subsets of a set $U$ of size $\lvert U\rvert=n$. Over all permutations $\pi$ of the set $\\{1,\ldots,m\\}$, I want to maximize the quantity \begin{equation} ...
7
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199 views

Does Dinur's proof of PCP Theorem imply a procedure for reconstructing a witness?

In Section 3.2 of On Syntactic versus Computational Views on Approximability by Khanna, et al., the authors state that an adaptation of the results from Proof Verification and Hardness of ...
12
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4answers
676 views

hardness of approximating the chromatic number in graphs with bounded degree

I am looking for hardness results on vertex coloring of graphs with bounded degree. Given a graph $G(V,E)$, we know that for any $\epsilon>0$, it's hard to approximate $\chi(G)$ within a factor ...
4
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0answers
121 views

Hardness of approximating |V|+(size of vertex cover)

I know that UGC implies a hardness of 2 for vertex cover, but is there a way to have this hardness on instances where the size of the vertex cover is at least $(1-\epsilon)\frac{|V|}2$? More generally ...
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2answers
239 views

Inappoximability status of Max One in Three SAT for satisfiable instances

What is the inapproximability status of Max-One-in-Three SAT for satisfiable instances?
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Hardness of approximation without the PCP theorem

An important application of the PCP theorem is that it yields "hardness of approximation" type results. In some relatively simpler cases one can prove such hardness without PCP. Is there, however, any ...
3
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2answers
177 views

What is the best-known inapproximability result for MIN-3CNF-DELETION?

I am really curious what the best-known inapproximability result is for MIN-3CNF-DELETION. To clarify, this is the problem of minimizing the number of unsatisfied clauses in a CNF SAT formula with at ...
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2answers
209 views

Approximability of the genus problem

What is currently known about the approximability of the genus problem? A preliminary search tells me that a constant factor approximation is trivial for sufficiently dense graphs, and an ...
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1answer
266 views

Is 3SAT problem APX-hard or not?

Could you point me a reference, an answer or it is an open question?
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260 views

What is meant by “if there exists a $\rho$-approximation algorithm with $\rho < 2$, then P = NP”?

For example, for the $k$-center problem we want to prove that a 2-approximation algorithm is optimal. A proof is presented on page 39 (Theorem 2.4) in Williamson and Shmoys, The Design of ...
21
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1answer
453 views

What is UG-hardness, and how is it different from NP-hardness based on the unique games conjecture?

There are many inapproximability results which rely on the unique games conjecture. For example, Assuming the unique games conjecture, it is NP-hard to approximate the maximum cut problem within ...
4
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1answer
197 views

Algorithms for Interval Coloring with Capacities and Demands

We are given a graph $G = (V,E)$, where each edge $e \in E$ has a capacity $c_e$. There are a set of $k$ requests $R_1,\ldots,R_k$. Request $R_i$ has a demand $d_i$, and has an interval $I_i = ...
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135 views

Is the dominating set problem constant-factor-approximable in undirected path graphs?

I am interested in the complexity of the minimum dominating set problem (MDSP) in some specific graph class. A graph is an undirected path graph if it is the vertex-intersection graph of a family of ...
4
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1answer
307 views

Why the reduction from MINIMUM SET COVER to MINIMUM DOMINATING SET means $c \log n$-inapproximability for MINIMUM DOMINATING SET

There is a well-known reduction from MINIMUM SET COVER to MINIMUM DOMINATING SET provided at http://en.wikipedia.org/wiki/Dominating_set#L-reductions (attributed there to Kann 1992, but seen, for ...
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391 views

Approximation results for 3-partition

The 3-partition as defined here is a strongly NP-complete decision problem. Consider one optimization problem of 3-partition where the $m$ subsets each have at most three elements and a sum of not ...
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1answer
883 views

What are good approximation algorithms for the subset sum problem so far?

By "good", I mean either the algorithm provides a relatively tight bound or it has a relatively fast running time. Any reference is welcome.
8
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1answer
211 views

Request for references on multicommodity flow-cut results

This is a somewhat subjective question. I am interested in studying the literature on multicommodity flow-cut results, especially the 'positive' results which show that flow is a good approximation to ...
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Inapproximability of multiterminal cut

In the multiterminal cut the input is a graph $G$ and a subset $T$ of its vertices. The task is to remove the minimum number of edges from $G$ such that there is no path connecting any distinct ...